Introduction to Path
Analysis
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Reminders about some important regression stuff
Boxes & Arrows – multivariate models beyond MR
Ways to “think about” path analysis
A bit about direct and indirect effects
A bit about mediation effects
measured vs. manifested  the “when” of variables
What path analysis can and can’t do for you…
About non-recursive cause in path models
Some ways to improve a path analysis model
The first multivariate model most of us learn is multiple regression
y’ = b1x1 + b2x2 + b3x3 + a
&
Zy’ = β 1Zx1 + β 2Zx2 + β 3Zx3
Things we study when learning this model …
Colinearity -- the difference between the information and
interpretation of bivariate r & b & or multivariate b
• how the amount and pattern of colinearity influences this
difference between bivariate r & b & or multivariate b
Suppressor effects – greater/differently signed multivariate
contribution that expected based on bivariate r & b
• colinearity patterns at their most interesting & informative
Things we learn when learning this model (cont.) …
Underspecification – our datasets/models don’t include all the
relevant predictors (causal agents)
• combined with colinearity & suppressor effects, this means
that the contribution of a predictor to “this model,” may not be
the same as its contribution to the “complete model"
Raw vs. Standardized weights -- the difference between b & β
• the greater comparability of β than b to compare the
relative unique contribution of predictors to the model
Causation -- the causal interpretability of the results of statistical
analysis is dependent upon the design used to collect the
data, not the statistical model used
• statistical control =/= experimental control
• “effect on” means “statistical relationship with” not “causal of”
One way to “think about” path analysis is as a way of “sorting out”
the colinearity patterns amongst the predictors – asking yourself
what may be the structure -- temporal and/or causal relationships
-- among these predictors that produces the pattern of colinearity.
“Structure” of a MR model – with
hypotheses about which predictors
will contribute
A proposed structure for the colinearity
among the predictors and how they
relate to the criterion – with hypotheses
about which paths will contribute
1
2
3
3
Crit
5
1
4
Crit
4
2
5
earlier
final cause
later
local cause
Alternative ways to “think about” path analysis…
• to capture the “causal paths” among the predictors and to the
criterion
• to capture the “temporal paths” among the predictors and to
the criterion
• to distinguish “direct” and “indirect” paths of relationship
• to investigate “mediation effects”
… to distinguish “direct” and “indirect” paths of relationship…
3
5
2 has a direct effect on Crit
1
4
Crit
2
• a “contributor” in both the regression
and the path models
• the regression model
5 does not have a direct effect on
Crit – but does have multiple
indirect effects
• not “contributing” in the regression
model could mistakenly lead us to
conclude it “5 doesn’t matter in
understanding Crit”
3
5
1
4
2
Crit
…to distinguish “direct” and “indirect” paths of relationship…, cont.
3
3 has a direct effect on Crit
5
1
4
Crit
2
3 also has an indirect effect
on Crit
• there’s more to the 3  Crit
relationship than was captured in
the regression model
3
5
1
4
2
Crit
… to investigate “mediation effects”…
Mediation effects and analyses highlight the difference between
bivariate and multivariate relationships between a variable and a
criterion (collinearity & suppressor effects).
For example…
For Teaching Quality & Exam Performance  r = .30, p = .01
• for binary regression β = r,
so we have the path model…
TQ
β=.3
EP
It occurs to one of the researchers that there just might be something else
besides Teaching Quality related to (influencing, even) Exam Performance.
• The researcher decides that Study Time (ST) might be such a variable.
• Thinking temporally/causally, the researcher considers that Study Time
“comes in between” Teaching and Testing.
• So the researcher builds a mediation model, getting the weights from a
multiple regression with TQ and ST as predictors of EP
… to investigate “mediation effects”…
The resulting model looks like …
β=.0
TQ
EP
β=.3
ST
β=.4
We might describe model as, “The effect of Teaching Quality on Exam
Performance (r=.30) is mediated by Study Time.”
We might describe the combination of the bivariate analysis and the multiple
regression from which the path coefficients were obtained as, “While Teaching
Quality has a bivariate relationship with Exam Performance (r=.30), it does not
contribute to a multiple regression model (β=.0) that also includes Study Time
(β=.40).
Either analysis reminds us that the bivariate contribution of a given predictor
might not “hold up” when we look at that relationship within a multivariate
model!
Notice that TQ is “still important” because it seems to have something to do
with study time – an indirect effect upon Exam Performance.
What path analysis can and can’t accomplish…
Cans
-- for a given structural model you can…
• evaluate the contribution of any path or combination of paths to
the overall fit of that structural model
• help identify sources of suppressor effects (indirect paths)
Can’ts
• non-recursive (bi-directional) models
• help decide among alternative structural models
• provide tests of causality (unless experimental data)
So… You have to convince yourself and your audience of the
“reasonableness” of your structural model (the placing of the
predictors), and then you can test hypotheses about which
arrows amongst the variables have unique contributions.
The “when” of variables and their place in the model …
When a variable is “measured”  when we collect the data:
• usually concurrent
• often postdictive (can be a problem – memory biases, etc.)
• sometimes predictive (hypothetical – can really be a problem)
When a variable is “manifested”  when the value of the
variable came into being
• when it “comes into being for that participant”
• may or may not be before the measure was taken
E.g., State vs. Trait anxiety
• trait anxiety is intended to be “characterological,” “long term”
and “context free”  earlier in model
• state anxiety is intended to be “short term” & “contextual”
 depends when it was measured
About non-recursive (bi-directional) models
1
Sometimes we want to consider
whether two things that “happen
sequentially” might have “iterative
causation” – so we want to put in
a back-and-forth arrow
Sometimes we want to consider
whether two things that “happen
at the same time” might have
“reciprocal causation” – so we
want to put in a sideways arrow
3
Crit
5
4
2
1
3
Crit
5
4
2
Neither of these can be “handled” by path analysis.
However, this isn’t really a problem because both are a
misrepresentation of the involved causal paths! The real way
to represent both of these is …
The things to remember are that:
1. “cause takes time” or “cause is not immediate”
• even the fastest chemical reactions take time
• behavioral causes take an appreciable amount of time
2. Something must “be” to “cause something else to be”
• a variable has to be manifested as an effect of some
cause before it can itself be the cause of another effect
• Cause comes before effect  not at the same time
When you put these ideas together, then both “sideways” and
“back-and-forth” arrows don’t make sense and are not an
appropriate portrayal of the causations being represented.
The causal path has to take these two ideas into account…
About non-recursive (bi-directional) models
1
If “5” causes “4”, then “4” changes
“5”, which changes “4” again, all
before the criterion is caused, we
need to represent that we have 2
“4s” and 2 “5s” in a hypothesized
sequence.
5
4
We also have to
decide when1, 2 & 3
enter into the model,
temporally &/or
causally. Say …
5
3
Crit
5
4
2
4
Crit
1
3
5
4
5
4
2
Crit
About non-recursive (bi-directional) models, cont…
1
When applying these ideas to
“sideways arrows” we need to
remember that the cause comes
before the effect.
3
Crit
5
4
2
To do that, we have to decide (& defend) which comes first – often
the hardest part) and then add in the second causation, etc.… As
well as sort out where the other variables fall temporally &/or
causally. Perhaps …
1
4
1
3
Crit
5
2
Some of the ways to improve a path analysis
TQ
For a given model,
consider these 4 things….
EP
ST
1. Antecedents to the current model
• Variables that “come before” or “cause” the variables in
the model
2. Effects of the current model
• Variables that “come after” or “are caused by” the variables
in the model
3. Intermediate causes
• Variables that “come in between” the current causes and
effects.
4. Non-linear variations of the model
• Curvilinear & interaction effects of & among the variables